Wigner D-matrix

The Wigner D-matrix is a square matrix, of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2j+1} , given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D^j_{m'm}(\alpha,\beta,\gamma) := \langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle = e^{-im'\alpha } d^j_{m'm}(\beta)e^{-i m\gamma} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha, \; \beta, } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma\;} are Euler angles, and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^j_{m'm}(\beta)} , known as Wigner's reduced d-matrix, is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcl} d^j_{m'm}(\beta) &=& \langle jm' |e^{-i\beta j_y} | jm \rangle\\ &=& [(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2} \sum_s \frac{(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!} \\ &&\times \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2s}\left(\sin\frac{\beta}{2}\right)^{m'-m+2s} \end{array} }

Relation with spherical harmonic functions

The D-matrix elements with second index equal to zero, are proportional to spherical harmonics (normalized to unity)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D^{\ell}_{m 0}(\alpha,\beta,\gamma)^* = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^m (\beta, \alpha )}

References

1. E. P. Wigner, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg Verlag, Braunschweig (1931).